An Inexact Interior-Point Lagrangian Decomposition Algorithm with Inexact Oracles

We combine the Lagrangian dual decomposition, barrier smoothing, path-following, and proximal Newton techniques to develop a new inexact interior-point Lagrangian decomposition method to solve a broad class of constrained composite convex optimization problems. Our method allows one to approximately solve the primal subproblems (called the slave problems), which leads to inexact oracles (i.e., inexact function value, gradient, and Hessian) of the smoothed dual problem (called the master problem). By appropriately controlling the inexact computation in both the slave and master problems, we can still establish a polynomial-time iteration complexity of our algorithm and recover primal solutions. We illustrate the performance of our method through two numerical examples and compare it with existing methods.